Several Jets players took a knee near the sidelines, while some formed a prayer circle as their team-mate was receiving medical attention.
“It was very sad, very emotional,” running back Breece Hall said. “You get on your knee and say a prayer for him and hope he’s OK.”
Glenn also praised the Jets’ medical staff for their handling of the situation.
The practice reportedly featured multiple fights between the two sides, with Stiggers having his helmet thrown down the field during a scuffle with Buccaneers’ Luke Goedeke.
Stiggers has 35 tackles, two fumble recoveries and three passes defended in 29 games for the New York outfit.
The Jets will host the Buccaneers in their NFL pre-season opener at MetLife Stadium, New Jersey on Friday (00:00 BST, Saturday).
BBC Sport has contacted the New York Jets for comment.